\documentclass[5pt]{article}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{proof}
\begin{document}

\infer[ DyR]{p_1 \vdash [b \bullet c]( \square_{A}p_5)}{
\infer[ \square R]{p_1^{b \bullet c} \vdash  \square_{A}p_5}{
\infer[ \bullet L]{(p_1^{b \bullet c})^{A} \vdash p_5}{
\infer[ ReArr']{(p_1^{b,c})^{A} \vdash p_5}{
\infer[ Stab(p_1,b,p_3)]{((p_1^{b})^{c})^{A} \vdash p_5}{
\infer[ Stab(p_3,c,p_5)]{((p_1^{b},p_3)^{c})^{A} \vdash p_5}{
\infer[ PrApp(A,p_5,p_5)]{((p_1^{b},p_3)^{c},p_5)^{A} \vdash p_5}{
\infer[Id]{p_5,((p_1^{b},p_3)^{c})^{A} \vdash p_5}{}}}}}}}}
\end{document}
